Which functions can't neural networks learn efficiently?
One of the important qualifications of the Universal approximation theorem is that the neural network approximation may be computationally infeasible.


  "A feedforward network with a single layer is sufficient to represent any function, but the layer may be infeasibly large and may fail to learn and generalize correctly." - Ian Goodfellow, DLB


I can't think of any function that I would definitively declare as unlearnable, but neural networks have many problems. Consider adversarial examples and adversarial patches, which highlight the poor generalization going on under the hood of recent advances in computer vision.

Neural Networks are also inherently limited by the innate priors baked into their architecture and the sample density of their training data. Check out this recent discussion at Stanford's AI Salon between Yann LeCun and Christopher Manning on innate priors if that is the kind of limitation you are talking about.